We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ ε} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/4 + ε} < m< n^{4/3}$. In this regime the independence number of $ G_{n,m}$ is concentrated on two values while the independence number of $ G_{n,p}$ is not; indeed, for $p$ in this regime variations in $ α( G_{n,p})$ are determined by variations in the number of edges in $ G_{n,p}$.